A Dynamic Risk Measure

The Iterated CTE – a Dynamic Risk Measure Mary R. Hardy Julia L. Wirch Dept of Statistics and Dept of Actuarial Mathematics Actuarial Science and Statistics University of Waterloo Heriot-Watt University Ontario, Canada Edinburgh, Scotland [email protected] [email protected] Abstract In this paper we present a method for defining a dynamic risk measure from a static risk measure, by backwards iteration. We apply the method to the CTE risk measure to construct a new, dynamic risk measure, the iterated CTE or ICTE. We show that the ICTE is coherent, consistent and relevant according to the defini- tions of Riedel (2003), and we derive formulae for the ICTE for the case where the loss process is lognormal. Finally, we demonstrate the practical implementation of the ICTE to an equity-linked insurance contract with maturity and death benefit guarantees. 1 1 Introduction 1.1 Quantile and CTE risk measures A risk measure maps a loss or profit random variable to the real line. The outcome is a quantification of the risk associated with a random loss. We commonly use risk measures in actuarial science to determine premiums or capital requirements for an uncertain future liability. The quantile risk measure is a traditional actuarial science measure, used both for capital requirements and for premiums. This has been separately adopted by the banking indus- try as ‘Value at Risk’ or VaR. The quantile risk measure with parameter ®, 0 · ® < 1 is the ®-quantile of the loss distribution. If the loss random variable is X > 0, then the ®-quantile random variable is denoted here by Q®(X); that is Pr[X · Q®(X)] = ® for a continuous loss random variable X. Using this measure for capital requirements is interpreted as setting capital requirements with a probability ® that the capital will be sufficient to meet the liability. Various authors have identified problems with the quantile approach. It is not coherent, in the sense of Artzner et al (2002); it can give perverse capital requirements, and it can be manipulated to mask substantial risks. It may encourage the sale of low frequency, high severity risks. Several authors have noted these. For further details of the theoretical and practical problems with the quantile or VaR approach see Artzner et al (1999), Boyle and Vorst (2003), and Wirch and Hardy (1999). The Conditional Tail Expectation (CTE) approach does not have these disadvantages. The CTE risk measure applied at some standard ®, where 0 · ® < 1 as before, for a random loss X which is continuous around the quantile value Q®(X), is C®(X) = E[XjX > Q®(X)] (1) That is, the capital required is sufficient to meet the loss in the event of the worst (1 ¡ ®) 2 event, on average. This equation for the CTE needs adjustment if Q®(X) falls in a probability mass. In the more general case the CTE with parameter ® is calculated as follows. Find ¯0 = maxf¯ : Q®(X) = Q¯(X)g (1 ¡ ¯0) E[XjX > Q ] + (¯0 ¡ ®) Q then C (X) = ® ® (2) ® 1 ¡ ® Another way of allowing for the probability mass problem is to use a distortion approach: 8 < y if 0 · y < 1 ¡ ® Let g(y) = 1¡® (3) : 1 if 1 ¡ ® · y · 1 and let SX (x) denote the decumulative distribution function of X, then Z 1 C®(X) = g(SX (x)) dx (4) 0 The CTE risk measure (also called Tail-VaR or expected shortfall) is coherent. It is also quite intuitive and is already widely used in actuarial science. The CTE risk measure has been adopted by the Canadian Institute of Actuaries in its recommendations with respect to the risk capital requirements for equity-linked contracts (SFTF (2002)), and is also widely used in the USA (see, for example, O’Connor (2002)). 1.2 The multi-period problem The quantile and CTE risk measures are static. That is, there is no explicit allowance for the risk measure to evolve as time progresses. A risk measure which is defined over a process rather than for a fixed liability is called a multi-period, or dynamic risk measure. The problem of dynamic or multi-period risk measures has been gaining much attention recently. Suppose an insurer has a liability due in 20 years. The static approach to risk measurement would give a value for the economic or regulatory capital to be held at the valuation date, 3 taking into consideration the distribution of outcomes for the future loss. However, the risk manager may be more immediately interested in having sufficient funds to meet the economic capital requirements in the following year, and the year after that and so on, as well as meeting the final liability. He or she will also be concerned not to hold excessive capital if the risk evolves favorably to the company. So a multi-period risk measure might allow not only the final liability but also for the intermediate capital requirements, taking into consideration how the loss process evolves. It should also allow for liability cashflows over time, as well as the single future liability. Artzner et al.(2003), Wang(2000) and Riedel(2003)have explored the multi-period problem using slightly different axiomatic approaches. The key to defining their risk measures is determining the appropriate closed convex set of test probabilities (scenarios) over which to calculate expected values. The test probabilities could be defined using a stopping-time process which will then allow any intermediate portfolio value to contribute to the risk measure calculation. Defining this set of test probabilities is the most difficult hurdle to implementing the multi-period risk measure. In this paper we take a more applied approach. Rather than start with axioms, we start with a single period risk measure; extend it to a multi-period framework, and then explore the characteristics of the dynamic risk measure. Our approach to the multi-period extension of the single period problem may be applied to any static risk measure. We focus on the CTE measure, as it is well known, coherent and commonly used for equity-linked insurance. In the risk measure literature there are commonly two different types of risk considered: ² The rolling horizon portfolio: in the application of Value at Risk to banking, typi- cally the risk is the potential portfolio loss over the following 10 days. Each day the risk is re-assessed with a new horizon. ² The fixed horizon portfolio; for example, where the risk measure is applied to a specific liability that expires at the fixed horizon. This would include the application of risk measures to a single equity-linked insurance contract, or a single cohort of 4 contracts. Where a whole portfolio of contracts with multiple maturity dates is to be risk measured, a fixed horizon (perhaps the final maturity) may be used, but a rolling horizon is also possible, particularly if new business is included. In this paper we are interested in the fixed horizon case, particularly as applied to the economic or regulatory capital for equity-linked contracts of a class which includes variable annuities and segregated funds. However, we believe that the method is easily adapted to the rolling horizon case. 1.3 Outline In the following section we use a simple binomial example to motivate the multi-period approach and to describe, intuitively, our solution. In Section 3 we define the iterated risk measure slightly more formally, and explore the characteristics of the measure. The rest of the paper considers two worked examples of the iterated CTE. In Section 4, we derive analytic results for a lognormal loss process. In Section 5 we work through an example for a segregated fund insurance contract1. 2 Example Consider the liability represented in the following 2-step model. 1Segregated fund contracts are known as Variable annuities in the USA and as Unit Linked contracts elsewhere. 5 ³³ 100 0:06³³ ³³ ³³ (1,1) P³ © PP p = 0:06©© PP © PP © PP 0 ©© 0:94 ©© (0) H HH HH ³³ 50 H 0:06³³ H ³³ HH ³³ 1 ¡ p = 0:94 (1,2) P³ PP PP PP 0:94 PP 0 At the start of the process, that is at node (0), the final liability X, due at time 2, takes values: 8 > <> 100 with probability 0:0036 X = 50 with probability 0:0564 > : 0 with probability 0:94 Suppose the regulator requires a 95% CTE to be demonstrated at each time point. At the start of the contract the 95% CTE of the final liability is: (0:0036)(100) + (0:05 ¡ 0:0036)(50) = 53:60 0:05 After one time unit the process is either at node (1,1) or node (1,2). At node (1,1), the 95% CTE to maturity is $100 . At node (1,2) the 95% CTE is $50. Note that the CTE held at time zero will be inadequate by $46.40 at time 1 if the process moves to node (1,1), which has 6% probability. So the static approach ignores a substantial probability of an additional draw on funds after the first time unit, despite the fact that the additional capital requirement is completely predictable in probability. For a multi-period risk measure we need a process Jt;T (X) where X is the future loss, t 6 is the time variable, T is the date at which the liability X matures. The simplest approach is to re-apply the single period risk measure at each time period. We call this approach the re-calculated CTE, since at each time unit we re-calculate the CTE to maturity. Another possibility, supposing the regulator is not involved, is simply to hold the initial risk measure capital of $53.60 through to maturity.

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